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Let $\mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert spaceequipped with a faithful normal semifinite trace $\tau$. A closed denselydefined operator $x$ affiliated with $\mathfrak{M}$ is called $\tau$-measurableif there exists a number $\lambda \geq 0$ such that $\tau\left(e^{|x|}(\lambda,\infty)\right)<\infty$. A number of useful inequalities,which are known for the trace on Hilbert space operators, are extended to traceon $\tau$-measurable operators. In particular, these inequalities implyClarkson inequalities for $n$-tuples of $\tau$-measurable operators. A generalparallelogram law for $\tau$-measurable operators are given as well.

Inequalities for trace on $\tau$-measurable operators